Saturday, March 11, 2006

Are Small State Voters Overrepresented in the Presidential Election?

[If you are allergic to statistics and probability theory, you may want to skip this post]

At first glance, it seems obvious that the answer is yes, since smaller states get more electoral vote relative to population than big states. But the question turns out to be a good deal more complicated than that.

Define a voter's voting power in an election as the probability that his vote will decide the outcome. Consider two states, one ten times the size of the other. Assume, temporarily, that the large state has ten times the electoral votes of the small. Assume, also temporarily, that the two states have the same probability distribution for the outcome, except that the distribution for the large state is proportionally stretched relative to that for the small. Assume that the chance that a state will determine the outcome of the presidential election is proportional to the number of electoral votes it casts.

On these assumptions, the analysis is straightforward. A voter in the small state has ten times as large a probability of deciding his state's outcome as a voter in the large state, but the small state has one tenth the probability of deciding the outcome of the election, so the two voters have the same voting power. Add into the model the fact that the small state has more than a tenth the electoral votes of the large and we get the obvious, and I think widely believed, conclusion—that small state voters are overrepresented.

There is, however, one more assumption we need to drop. The probability distribution for the outcome in the large state is not simply that in the small state stretched out by a factor of ten. The law of large numbers tells us that, all else being equal, the distribution in the large state will be more sharply peaked than in the small. If the peak of the distribution is at .5, that means that the probability of a one vote victory, making every voter on the winning side decisive—changing his vote would reverse the outcome—is more than a tenth as high in the larger state as in the smaller. That gives us the opposite of the previous result: Voters in smaller states are underrepresented. Since the two effects go in opposite directions, one cannot tell, on theoretical grounds, what the net effect is.

There is, however, one more complication we need to deal with. In the previous paragraph, I assumed that the peak of the probability distribution for a state's electoral outcome was at .5—that the random voter had exactly .5 probability of voting each way. That is unlikely to be true, even if we limit ourselves, as we should, to states where the vote will be close.

Suppose the typical voter has a .51 probability of voting Republican. The sharper probability distribution for the larger state increases the probability that the outcome will be 51/49. But it might, depending on how sharp the peak is, decrease the probability that the outcome will be 50/50, and so the probability that one vote will be decisive.

My conclusion is doubly indeterminate. One factor results in overrepresenting voters in small states. The other might result in over or underrepresenting them. Figuring out the actual effect would require a fairly careful examination of detailed electoral evidence. Being lazy—hence a theorist—I will leave that job to someone else.

15 Comments:

I know of at least some people who have worked on it (a postdoc at my college worked with some students here, and one's doing a thesis on it). Link to the paper here. The summary:

"Many believe that the Electoral College over-represents voters in less populous states since it guarantees each state at least three electoral votes. This is incorrect! In fact, a probabilistic analysis shows that the Electoral College exhibits an inherent bias favoring voters in more populous states. Indeed, a voter in California is 3.4 times more likely to change the outcome of a Presidential election with her single vote than a voter in Montana...We prove that it is impossible for the Electoral College, or almost any federal weighted voting system, to endow each voter with exactly the same power."

On the other hand, since voters in the same state may share many interests (and since the odds of any one voter determining the election results are miniscule), it may be more instructive to look at the voting power of individual states, in which case the small states have disproportionate influence.

Not that this necessarily means that the Electoral college, or at least some federally weighted voting system, is a bad thing; it inter alia requires a candidate to have broad acceptance as well as deep. Though it makes some sense to adjust the electoral college to weigh more fairly, as Elgin explores, the idea of a weighting system has some merit.

My thanks to jadagul for the link. I can't tell from the webbed material just what the methodology was, or whether the "probabilistic analysis" took account of the issues I raised in my post, so I have emailed the author asking to see the paper.

Does this imply that a third party candidate is more likely to affect the outcome of an election than his popular vote percentage would indicate?

Assuming that his vote percentage is constant for most states, he would be more likely to alter the electoral vote in the largest states, with a resulting higher probability of changing the outcome of the election.

A much simpler analysis technique: Look at where the presidential candidates campaigned. My impression is that small states get a far smaller share of attention from national campaigns than their proportion of the population. Whatever one may think of these guys' ability to run the country, they wouldn't be the major party candidates if they were politically incompetent...

If ever a count showed a margin of one vote, there would be at least one recount, which would almost certainly show a different result. Then, there would certainly be a court challenge, such challenge almost certain to be decided on some basis other than the one-vote differential in one of the recounts, i.e., on the basis of real political power.

When I multiply these two near-certainties by the near-impossibility of a one-vote margin, I get P=0. Maybe someone else wishes to claim that there is an epsilon there, but I'm not buying it.

In other words, I agree with Mark - no difference between probabilities for large- vs. small-state voters.

jason: But what if instead of being the one vote that gives a candidate more votes than his opponent, you're the one vote that gives a candidate a sufficiently large margin of victory to avoid sending the issue to the courts?

FXKLM: I realize we're getting into Sorites Paradox territory here, but do you think there's any point where one vote makes the difference between a legal battle and no legal battle? Even if the law says something like "we recount if and only if the margin is less than or equal to 1000 votes," if I lost by 1001 votes (and were a self-interested politician who actually wanted the office), you can bet I'd be suing to try to get two ballots thrown out. So we're back to the courts.

ATR: Tom was definitely using the Banzhaf power index; he also said as much in his online materials (and it's quite close to the methods described in the main post), so I assume Professor Friedman wanted more details that weren't online.

With a large enough victory, there is no legal challenge. With a small victory, there is a legal challenge. We may not be able to determine the specific number of votes where that line is crossed, but it has to exist somewhere.

The interesting thing about this interpretation is that it's possible to cast the deciding vote and not know it even after the votes are tallied.

FXKLM - It did occur to me, only it was while I was brushing my teeth after making the comment! To some extent I think there is a valid point there, but probably not enough to claim that P is definitely greater than zero, because...

If we are (I think rightly) taking the issues of court challenges and what I vaguely called "real political power" seriously, then there's yet another elephant in the room we should take seriously, and it's election fraud -- counting of nonexistent ballots, disposal of uncounted valid ballots, dishonest voter-eligibility challenges, etc. Forget knowing if your vote was decisive; wonder instead whether your ballot was counted.

Obviously ballot fraud goes hand-in-hand with considerations of "real political power", but consider this: if you're powerful enough to swing a court challenge your way, you may well be powerful enough to swing the initial ballot count into the range necessary to claim the need for a recount or a court case. And here I will assert without citation that this general scenario has played out in electoral reality for many an overly democratized country.

You know what's also interesting about this discussion: We could argue that, even if a one-vote margin were ever observed, the "decisive" voter did not actually swing the election, but that the winning side would have won anyway through a recount or court challenge that proved unnecessary given the exact result.